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Low Noise Measurements François D. Parmentier Ecole Meso 2016

Low Noise Measurements (for mesoscopic physics, quantum transport & circuits)

François D. Parmentier Service de Physique de l’État Condensé, CNRS-CEA Saclay

[email protected]

‘a gentle journey in a world of many body problems & soldering irons’


Why this course?

1. They asked me…

2. low noise measurements for experimentalists

3. low noise measurements for theorists?

Low Noise Measurements



Martinis, Devoret, Clarke, PRB 35, 4682 (1987)


(incomplete) Bibliography

• Books: Lounasmaa – Experimental principles and methods below 1K (1974) Pobell – Matter and Methods at Low Temperatures (1992) Horowitz & Hill – The Art of Electronics (1989) White – Experimental Techniques in Low-Temperature Physics (1979) Ventura & Risegari – The Art of Cryogenics (2008)

• PhD theses: H. Pothier (1991) B. Huard (2006) J. Gabelli (2006) H. le Sueur (2007) F. D. Parmentier (2010) S. Jezouin (2014) T. Jullien (2014)

• review papers: F. Giazotto et al., Rev. Mod. Phys., 78, 217 (2006) D.C. Glattli, Eur. Phys. J. Special Topics 172, 163–179 (2009) A.A. Clerk et al., Rev. Mod. Phys. 82, 1155 (2010)

+ references mentioned in the slides (and references therein)



Landauer Buttiker formalism: electron quantum transport = coherent conductor (≡scattering matrix 𝒮) + ideal reservoirs

kBT1, µ1 k2T2, µ2

reservoir 1

𝒮

reservoir 2

• how does one realize ideal reservoirs, with well controlled kBT & µ?

• how does one measure 𝒮 precisely in both linear & non-linear regimes? transmission=conductance ∝ 𝑡 2

𝒮 =𝑟 𝑡𝑡 −𝑟


Energy scales in mesoscopic transport

“intrinsic”: - quantum dots: charging energy Ec / level spacing Δ - Kondo temperature kBTK

- superconducting gap Δ / Andreev bound states energy EA

- …

“external probes”: - dc/ac voltage Vdc/Vac - e-mag field / photons ħω - temperature kBT - …

need to be tunable from << Ec , Δ , … to ≥ Ec , Δ , …

APL 73, 2992 - 2994 (1998)

STM tunneling conductance Nb / Au

Δ

1 K ⟺ 86 µV ⟺ 20 GHz


Example: transport in a quantum dot

IQD VD

VG CG

resonance width affected by temperature

Altimiras et al., Nat. Phys. 6, 34 - 39 (2010)

휀

f(휀)


Outline

1. low temperature experiments • cryogenic systems • lattice vs electron temperature • filtering & shielding

2. low noise cryoelectronics • signal vs noise • lock-in measurements • measurement configurations

3. beyond dc conductance • microwave measurements • shot noise


pumped liqu. 4He

4.2 K 2.17 K 1 K 0.87 K 0.3 K 0.01 K

superfl. 4He 4He / 3He

phase separation

pumped liqu. 3He

4He / 3He dilution refrigerators

cern

Aalto U.

He-based cryogenic systems Part 1 low T exps.


4He / 3He dilution refrigerators

• circulated 4He / 3He mixture • continuous operation down to 1 mK • needs 4He bath + cooling power at ~1 K (1 K pot or Joule-Thomson expansion)

Oxford

Part 1 low T exps.


Wet vs dry dilution refrigerators

‘Wet’ fridge: dilution unit dipped in liquid 4He bath

Das et al., Low Temperature Physics LT9, 1253-1255 (1965)

+ reliable + (relatively) fast cooldowns - liquid 4He consumption expensive - requires regular refills (week-end…) - limited experimental space - vacuum isolation = low temperature seal - cryogenic liquids hazards

Part 1 low T exps.


Wet vs dry dilution refrigerators

‘dry’ fridge: precooling down to 4 K by mechanical refrigerator (compression-expansion cycles in “pulse-tubes”)

+ automatized & autonomous + no more liquid 4He transfers! + huge experimental space - mechanical vibrations - high electricity & cooling water consumption - VERY POOR efficiency (5 kW electrical power → <1 W cooling @ 4K) - sensitive to electrical failure

6 mK

100 mK

800 mK

3 K

60 K

pu

lse-tub

e refrigerator

Oxford

game-changer for labwork / labs funding / cryogenics industry

Part 1 low T exps.


So you got a fridge… then what?

peak width ∝ sample temperature Tel

∝𝜕𝑓Fermi(휀)

𝜕휀

100

100 200 300

200

300

T el (

mK

)

Tfridge (mK)

discrepancy Tel/Tfridge at low temperature?

Part 1 low T exps.


Electron vs lattice temperature

𝑄 e−ph = Σ𝑈(𝑇el5 − 𝑇wire

5 )

𝑄 ph−ph = 𝐾𝐴(𝑇wire4 − 𝑇fridge

4 )

𝑄 el = 𝐿0 (𝑇wire2 − 𝑇fridge

2 )/𝑅GND

metallic wire

Tel Twire

Tfridge cold bath

RGND

𝑄 𝑖𝑛

electrons, Tel

wire phonons, Twire

cold bath, Tfridge

𝑄 e−ph

𝑄 ph−ph

𝑄 el

𝑄 𝑖𝑛

thermal transport model:

electron-phonon cooling:

phonon-phonon cooling (Kapitza resistance):

electronic heat transport (Wiedemann-Franz law):

Σ: coupling constant (depends on material) 𝑈: wire volume

𝐾: coupling constant (depends on materials) 𝐴: contact area

𝐿0 = 𝜋2𝑘𝐵2/3𝑒2: Lorenz number

Rev. Mod. Phys., 78, 217 (2006)

Part 1 low T exps.


Example: metal contact on meso. conductor

electrons, Tel

wire phonons, Twire

cold bath, Tfridge

𝑄 e−ph

𝑄 ph−ph

𝑄 el

𝑄 𝑖𝑛 Tel

Twire

Tfridge =10 mK

cold ground / Si substrate

RQ = h/e²

𝑄 𝑖𝑛

300 µm x 300 µm x 100 nm Cu contact Σcu=2x109 Wm-3K-5

𝐾cu/Si=166 Wm-2K-4

10

15

20

25

1 10 100

Q in (fW)

T (mK)

1 10 100

Q in (fW)

1

10-2

102

Q el/ph (fW

)

minimize power 𝑄 𝑖𝑛 incoming on sample

Part 1 low T exps.


Example: metal contact on meso. conductor


RQ = h/e²

10 mK

4 K

300 K

𝑄 𝑖𝑛

𝑄 𝑖𝑛 : • thermal conduction from warmer parts • photons (blackbody radiation)

• Joule power dissipated in the contact

wire thermalization

shielding & filtering

small electrical signals

Part 1 low T exps.


Wire thermalization thermal conductivity 𝜅 @ low temp

10 mK

4 K

300 K metals: 𝜅 ∝ 𝑇

heat mainly carried by electrons (Wiedemann-Franz law):

𝑄 el = 𝐿0 (𝑇hi2 − 𝑇lo

2 )/𝑅wire

Thi

Tlo

use resistive wire

Lounasmaa - Experimental principles and methods below 1K (1974)

Part 1 low T exps.


Wire thermalization

cold dielectric substrate

Ti =1 K Tf =10 mK

10 mK L

wire length L to thermalize electrons from 1 K to 10 mK ?

hyp.: electrons well thermalized to wire phonons

Twire(x)

cold bath, 10 mK

Twire(x+dx) Twire(x-dx)

W-F W-F … …

Kapitza 𝐿 =2 𝐿0𝜎

𝐾

𝑑𝑇

𝑇4 − 𝑇𝑓4

𝑇𝑖

𝑇𝑓

Twire (K)

L (m)

electrical conductivity

wire material is crucial!

Part 1 low T exps.


Photons impinging on the sample

• TE/TM photons propagating in the wires

• TEM photons modes of the sample box


10 mK

4 K

300 K

TEM photons

TE+TM photons

sample box

Part 1 low T exps.

shielding!


non-thermalized photons

• radiative heat transfer: Stefan-Boltzmann law

T (K)

P A (

W/m

²)

𝑃𝐴 W m² =2𝜋5𝑘𝐵

4

15ℎ3𝑐2𝑇photons4

• electromagnetic noise on the gates

Part 1 low T exps.


non-thermalized photons II • charge transfer in coherent conductor coupled to electromag. envt:

Dynamical Coulomb Blockade

VD

VG CG Z

env

tunneling rate Γ affected by prob. 𝑃 𝐸 for envt to absorb photon with energy 𝐸 :

Γ 휀 = 𝑑𝐸 Γ0 휀 − 𝐸 𝑃(𝐸)

𝑃 𝐸 ≈1

2𝜋ℏ 𝑑𝑡𝑒

𝑖𝐸𝑡ℏ × exp

2𝜋

ℏ𝑅𝐾 𝑑𝜔

𝑆𝑉(𝜔)

𝜔2(cos𝜔𝑡 − 1) 𝑍env ≪ 𝑅𝐾 →

𝑆𝑉(𝜔) ∝Re[𝑍env]ℏ𝜔

exp ℏ𝜔 𝑘𝐵𝑇ph − 1 (Planck’s law)

Devoret et al., Phys. Rev. Lett. 64, 1824-7 (1990) Nazarov & Ingold, Single Charge Tunneling (1992) Martinis & Nahum, PRB 48, 18316 (1993)

Part 1 low T exps.


[…]

non-thermalized photons

…

D. C. Glattli, P. Jacques, A. Kumar, P. Pari, & L. Saminadayar, Journal of Applied Physics 81, 7350 (1997)

Martinis & Nahum, PRB 48, 18316 (1993)

Part 1 low T exps.


Filtering, wiring and shielding

6 mK

100 mK

800 mK

3 K

60 K

Z. Iftikhar et al., Nature Communications 7, 12908 (2016)

Part 1 low T exps.


Filtering: coax. wires

resistive wires needed for electron thermalisation just add capacitance!

cryo coaxial wire: lossy transmission line

outer conductor (ground) • minimize heat load on fridge • define a good ground • thermal anchor to fridge → thin and slightly resistive

inner conductor (signal) • electron thermalization • suppress thermal photons → resistive • minimize heat load → thin

dielectric • defines good capacitance • good thermal transfer between inner & outer conductors → thin

typ. R=50-100 Ω/m C=100 pF/m

• commercial cryo coax • thin resistive wire threaded in small CuNi tube

Part 1 low T exps.


copper clamp

Cu lines on kapton

coax Thesis J. Gabelli (2006)

• planar copper lines: not good below 1 K!

Filtering: discrete elements

Thesis H. Pothier (1991)

Part 1 low T exps.


Outline

1. low temperature experiments • cryogenic systems • lattice vs electron temperature • filtering

2. low noise cryoelectronics • signal vs noise • DC & AC meast, lock-in • measurement configurations

3. beyond dc conductance • microwave measurements • noise


Low noise cryoelectronics

measurement of GDOT(VD,VG) at low temperature ?

electrons are cold photons are filtered out

VG VD

300 K

Tel=10 mK

Part 2 low noise cryoelec.


Signal vs noise

VG

GDOT signal

noise

real measurements are always noisy:

Maximize signal / noise ratio (SNR): A) noise sources B) which signals?

+ + - VG VD

300 K

Tel=10 mK



Noise sources

• thermal noise • 1/f noise • amplification noise • parasitics, room T:

o flux o ground loops

• parasitics, cryogenics o thermovoltages o triboelectrics o microphonics



Thermal (Johnson-Nyquist) noise

Johnson, Phys. Rev. 32, 97 (1928) (experiment) Nyquist, Phys. Rev. 32, 110 (1928) (theory)

thermal agitation of charge carriers in a conductor (resistance R, temperature T)

voltage fluctuations 𝑣² = 4𝑅𝑘𝐵𝑇 units: V²/Hz (or V/√Hz )

𝑅, 𝑇 = 0 𝑣² = 4𝑅𝑘𝐵𝑇 𝑅, 𝑇

Electrical example of fluctuation-dissipation theorem!

good primary thermometer for low (and high!) temperatures (also a good way to calibrate amps) N

ois

e (V

²/H

z)

T (K)

0.1 0.2 0.3 0.4

50 Ω @ 300 K → 0.9 nV/√Hz



1/f (flicker) noise sample in electrostatic environment:

surrounding trapped charges = fluctuating two-level systems

low frequency fluctuations!

log(f)

log(noise)

thermal noise

very unfavorable for DC measurements, thermally activated

Fc ≈1-100 Hz



Amplification noise: field-effect transistors

field-effect transistor (FET)

Drain Source

channel

Vbias

Vg+δVg

VD+GδVg

𝐺 ≈ −𝑔m𝑔c

Main noise sources: - 1/f noise - thermal noise of the channel - shot noise of the gate leakage current

Bordoni et al., Rev. Sci. Instrum. 52, 1079 (1981)

𝑔m =𝜕𝐼channel

𝜕𝑉𝑔

𝑔c =𝜕𝐼channel

𝜕𝑉𝐷

Ichannel

transconductance

channel conductance

typical noise: 𝜹𝐯²~𝟏 𝐧𝐕/ 𝐇𝐳


sources independent of measured impedance Rm: voltage noise sources dependent of Rm: current noise

Drain

Source

Gate

Rm


Room T: flux & ground loops

Fluctuating magnetic flux (50 Hz, …) in the loop induces noise

get rid of loops!

VG VD

300 K

Tel=10 mK

+ + -

Φ




get rid of loops:

• twisted pairs • shielding

VG VD

300 K

Tel=10 mK

+ + -




get rid of loops:


Tel=10 mK

+ + -

Φ

IGND

IGND

IGND

beware of ground loops! long, poorly conducting GND wires

parasitic fluctuating IGND

pick up flux

noise!!

VG VD




get rid of loops:


beware of ground loops!

• single well defined GND (close to fridge) • short, low resistance wires to the GND

VD

Tel=10 mK

+ + -

VG

GND



Thermoelectric voltages

6 mK

100 mK

800 mK

3 K

60 K

Tel=300 K

Tel=10 mK

Seebeck effect: thermoelectric dc voltage between 300 K and 10 mK

𝑉therm = S∆𝑇 (∆𝑇

𝑇≪ 1)

𝑉therm~1 − 100 µ𝑉

slow fluctuations with temperature!

unfavorable for DC measurements



DC vs finite frequency meast VD=0 : GDOT=0 (no signal from sample)

1/f noise + thermovoltage

thermal+ amp noise

Gm

es (

a.u

.)

PhD student working hours

FFT

freq. (number of thesis per decade?)

ℱ[G

mes

] (a

.u.)

thermal noise

log-log scale

+ + -

VG VD

Tel=10 mK



DC vs finite frequency meast after working ~500 hrs, student A remembers to turn VD on.

VD=1 V: GDOT=1 a.u.

Gm

es (

a.u

.)


VD on

ΔGmes=1±1…

DC G-meter

• ‘instant’ DC meast → very noisy! • averaged DC meast → drift!!

+ + -

VG VD

Tel=10 mK



DC vs finite frequency meast Solution: turn VD on periodically

AC G-meter

VD=0,1,0,… V → GDOT=0,1,0,… a.u.

Gm

es (

a.u

.)


VD on

VD off

FFT


ℱ[G

mes

] (a

.u.)

log-log scale

VD=0

VD=0,1,0,…

f0 harmonics

+ + -

VG

VD

Tel=10 mK



DC vs finite frequency meast Solution: turn VD on periodically + measure power at f0

VD=0,1,0,… V → GDOT=0,1,0,… a.u.


ℱ[G

mes

] (a

.u.)

log-log scale

VD=0

VD=0,1,0,…

f0 harmonics

spectrum analyzer

Gm

es (

a.u

.)

GDOT (a.u.) / VD (V)

noise floor

AC meast: less noise

+ + -

VG

VD

Tel=10 mK



Lock-in measurement turn VD on periodically + multiply output signal by sine wave at same period

Gm

es (

a.u

.)

GDOT (a.u.) / VD (V) Standard meast tool in exp. physics

sin(f0t+φ)

Gmes[cos(φ)-cos(2f0t+φ)]

http://www.thinksrs.com/

low pass

Gmescos(φ)

lower noise floor (same avg time)

noise floor

integ. power

lock-in

Lock-in amplifier

+ + -

VG

VD

Tel=10 mK



Differential conductance meast

lock-in: great for small AC signals detection

drive sample with small (linear) AC signal ∂V on top of DC signal

VG VD

Tel=10 mK

∂VD

IQD+ ∂IQD

+ lock-in measurement of diff. conductance gDOT(VD)=∂IQD/∂VD

• proportional to D.O.S.! • access to the linear (VD=0) regime • emphasizes non-linear features

APL 73, 2992 - 2994 (1998)

STM tunneling conductance Nb / Au








Dynamical Coulomb Blockade in ultrasmall tunnel junctions

Delsing et al., PRL 63, 1180 (1989)

VG VD

Tel=10 mK

∂VD

IQD+ ∂IQD








eVac<kBTel

VD

Sample (Tel)

Vac

1 µV ↔ 11.5 mK

AC voltage scale set by Tel

Kondo effect in semiconductor quantum dots

Cronenwett et al., Science 280, 540 (1998)

45 mK

270 mK



Lock-in meast configurations

• differential conductance ∂IQD/∂VD

VG VD

∂VG

• transconductance ∂IQD/∂VG

straightforward , quantitative extraction of GDOT / 𝒮-matrix

out-of-equilibrium meast: VD1,2,… applied to various terminals

(requires calibration of lever arm VG ↔εDOT)

VG VD

∂VD

IQD+ ∂IQD

IQD+ ∂IQD



Meast configs.: 2/4 points

2 points resistance/conductance meast 4 points resistance/conductance meast

Sample

I so

urc

e

V

I so

urc

e

Sample

V

2 points: V/I=Rsample+2raccess

V

Isource

current source: Rint ≪ Rsample volt-meter: Rint ≫ Rsample

raccess r acc

ess

raccess= wires, contacts, …

‘false’ reading of Rsample

Isource

4 points: V/I=Rsample

be careful when measuring very high Rsample!

no contribution from raccess



4 pts & more: non-local meast

A) unveils non-local effects

drive current in one side of sample & measure non-local voltage on other side

I so

urc

e

Sample

V

I so

urc

e

V

Φ

Umbach et al., APL 50, 1289 (1987)

B) probes chirality: quantum Hall effect

I so

urc

e

Vxx

VH

B



Even more complex (chiral) geometries

complex quantum circuits: → several simultaneous lock-in measurements at various frequencies!

quantum dot 1 quantum dot 2

Iac1 @ f1

Iac2 @ f2

τdot2 (reflection)

τdot1 τdot2 (total transmission)

τdot1 (reflection)

use chirality and lock-in measurement to independently extract all parameters

R. Rodriguez’ PhD experiment, SPEC



Conclusions of parts 1 & 2

1) electron thermalization & filtering crucial for low T quantum transport experiments

H. le Sueur, PhD thesis (2008)

2) small signals in cryo envt: - beware of noise sources & signal/GND loops

- lock-in meast: how large is Vac?


Outline

1. low temperature experiments • cryogenic systems • lattice vs electron temperature • filtering

2. low noise cryoelectronics • signal vs noise • DC & AC meast, lock-in • measurement configurations

3. beyond dc conductance • microwave measurements • noise


Microvave signals in meso. physics

ℎ𝜈

photon-assisted tunneling

control & spectroscopy: circuit quantum electrodynamics

Schoelkopf & Girvin, Nature 2008

open quantum system dispersive readout

Delbecq et al., PRL 2011

time dependent transport & quantum capacitances

Gabelli et al., Science 2006

Part 3 beyond dc


Microvave signals in meso. physics Part 3 beyond dc

typical frequencies ℎ𝑓 > 𝑘𝐵𝑇el 𝑓 > 1 GHz

can’t use usual filtered dc lines !

solution: differentiate input and output lines


Microvave signals in meso. physics Part 3 beyond dc

solution: differentiate input and output lines

input: use attenuators

=

r1

r1

r2

• attenuates (yes!) µwave power • res. bridge: electron thermalization + decoupling • reduces multiple reflections

sample

300 K

10 mK

signal analyzer

output: use circulators

induces non-reciprocity in µwaves propagation (e.g. using ferrites)

1

2

3 =0 0 11 0 00 1 0

BUT: bulky + limited bandwidth


microwaves: impedance matching Part 3 beyond dc

load Zl

transmission line Zc

µwave signal

wave propagation:

reflection coefficient on the load 𝑟 =𝑍𝑙−𝑍𝑐

𝑍𝑙+𝑍𝑐

impedance matching: 𝑟 = 0 for 𝑍𝑙 = 𝑍𝑐

problem: • Z=50 Ω for most commercial µ-wave electronics • but Z≈RK=25813 Ω for typical meso. circuits…

hard to have good coupling & large bandwidth!

transmission line

50

Ω

samp

le

shunt sample with 50 Ω load: poor coupling, good bandwidth

transmission line

samp

le

‘stub tuner’: good coupling, small bandwidth T. Hasler, et al., Phys. Rev. Applied 4, 054002 (2015)

op

en lin

e


(complicated) example: double balanced amplifier Part 3 beyond dc

µwave Mach Zehnder interferometer

U1 U2

hybrid coupler = beam splitter

F.D. Parmentier et al., Rev. Sci. Inst. 82, 013904 (2011)

U1 U2


(quantum) noise measurements

Part 3 beyond dc


shot noise Part 3 beyond dc

shot noise: granularity of charge transfers + partitioning

𝑆𝐼𝐼 = 2𝑒𝐼(1 − 𝐷)

Fano factor


zero & finite frequency quantum noise Part 3 beyond dc

SII(ω)

ω 0

absorption noise

emission noise

A.A. Clerk et al., Rev. Mod. Phys. 82, 1155 (2010).

current correlator: 𝑆𝐼𝐼 𝜔 = 𝑑𝑡 𝐼 (𝑡)𝐼 (0) 𝑒𝑖𝜔𝑡

noise meast = measuring a power over some freq. bandwidth


very (very very) small signals Part 3 beyond dc

𝑆𝐼𝐼 ≈ 2𝑒𝐼(1 − D)

𝑆𝐼𝐼 ≾ 2𝑒 × 1 nA ∼ 10−30 −10−28 A2/Hz

QPC, 1 channel, D=0.5 T=6 mK

𝑆 𝐼𝐼

Z. Iftikhar et al., Nature Communications 7, 12908 (2016)


very (very very) small signals Part 3 beyond dc

samp

le

𝑆𝐼𝐼 ≈ 2𝑒𝐼(1 − D)

𝑆𝐼𝐼 ≾ 2𝑒 × 1 nA ∼ 10−30 −10−28 A2/Hz

Rm

I sou

rce

𝛿v²~1 nV/ Hz

signal to noise ratio:

SNR =ℛ2𝑆𝐼𝐼𝛿v²

1

𝑁mes

≈ 10−10ℛ2

𝑁mes

𝑁mes = ∆𝑓 × 𝑡mes

ℛ

needs a long averaging time!


noise meast with tank circuits Part 3 beyond dc

T

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